Distance from arbitrary shape so we can treat it as a point charge

[oddjobmj]

Homework Statement

Non-uniform charge distribution over a randomly shaped object. This object will fit inside a sphere centered on the origin with radius r. What is the minimum distance from the origin that we can assume such that we can treat the electric field as if it were generated by a point charge at the origin?

Homework Equations

Gauss' law?

[itex]\vec{E}[/itex]=k[itex]\frac{q}{R^2}[/itex][itex]\hat{R}[/itex]

The Attempt at a Solution

My first guess is Gauss' law because it works with arbitrary shapes. As long as we can ensure that our point of interest is outside our shape the field (or the flux leaving the surface, at least...) should be equivalent to one from a point charge. In other words as long as R>r where R=point's distance from the origin. This is new material to me though so I am struggling a little to wrap my head around this and relating flux and field and the shape/symmetry is throwing me off. I'm questioning this assumption now:

My professor recently noted that we should consider the worst case scenario (maximum divergence from a point) and implied some (simple) calculations are involved. I believe the 'worst case' scenario would be either 1) where the shape is actually a sphere with radius r or 2) it is a point charge somewhere besides the origin.

If it is a point charge on, say, the x-axis where x=r AND we calculate the field at x=-r while assuming that the point charge is at the origin our estimation will be off by a factor of 4 since R is squared and our assumed distance is one half the actual.

Any suggestions? I really want to make sure I understand this because I believe it is pretty fundamental but apparently it's giving our class quite a bit of trouble.

 

[Orodruin]

The worst case would be two point charges with charge Q+q and -Q, respectively, where Q >> q, located in x = r and x = -r respectively. In addition to the change in the monopole field, this will also give you a large dipole component which falls off as 1/r^3.

The case of the spherical shell is not a worst case as the field outside the sphere shows the exact same spherical symmetry as that of a point charge.

It is not really possible to define a radius where the field is dominated by the monopole component as it will involve some arbitrariness in deciding what "dominated" means. R>r would typically not be enough to make the field a monopole field.

 

[oddjobmj]

Thank you for your input, Orodruin!

Well, I will say that the professor made it clear several times (and it occurs in text right before this problem) that the acceptable margin of error for our answers in the class is 1%. I believe that this answer has to be within 1% which should allow me to solve for the minimum distance.

Given two point charges at x=r and x=-r, as you noted, I should calculate the field at some point on, say, the y-axis?

Also, I am a little confused about the magnitudes you chose for those charges. To be clear, Q1 and Q2 are very nearly equal in magnitude but also opposite in sign. They differ in magnitude by q? Why not Q1=-Q2?

Edit: Also, it does state that our object is actually a single charged mass. It does have a non-uniform distribution though so I guess it wouldn't be too big of a stretch to distribute the charges in the above fashion.

 

[Orodruin]

Also, I am a little confused about the magnitudes you chose for those charges. To be clear, Q1 and Q2 are very nearly equal in magnitude but also opposite in sign. They differ in magnitude by q? Why not Q1=-Q2?

If Q1 = -Q2 the total charge in the volume is zero and not q. Regardless of what you do, you can never find an R that will be large enough if you allow arbitrarily large charges Q.

 

[HalfLostAndSearching]

I would first clarify the scenario and assumptions being made. Is the non-uniform charge distribution over the randomly shaped object known or unknown? Are we assuming the object is a conductor or an insulator? These details can greatly affect the analysis and result.

Assuming the object is a conductor and the charge distribution is known, we can use Gauss' law to calculate the electric field at a point outside the object. However, if the charge distribution is unknown or the object is an insulator, we may need to use other methods such as integration to determine the electric field.

In either case, the minimum distance from the origin that we can assume for treating the electric field as if it were generated by a point charge would depend on the shape and charge distribution of the object. It may not necessarily be related to the radius of the sphere containing the object.

Furthermore, the concept of "worst case scenario" may not always apply. For example, if the object is a sphere with radius r, the electric field at a point outside the sphere will still be given by Gauss' law with the charge distribution integrated over the entire sphere. In this case, we cannot simply assume the point charge is located at the center of the sphere.

In conclusion, the minimum distance from the origin that we can assume for treating the electric field as if it were generated by a point charge will depend on the specific scenario and assumptions being made. Careful consideration and analysis is needed to accurately determine this distance.